『愛媛大学社会共創学部紀要』第７巻第２号

論2023年

p.13-19

説Growth model of forests as a basis for climate change mitigation andnature-based solutions: Tentative estimation of a Japanese cider forest’s
growth model in Ehime
Noriko IRIE（Environmental Design）
Kumiko TAKESHIMA（Regional Resource Management）
Naoko KAWAHARA（Kindai University）
入竹
川江
島原

賀 子（環境デザイン学科）
久美子（地域資源マネジメント学科）
尚 子（近畿大学経営学部）

Keywords: forest, Ehime ciders, sustainable usage, growth model, parameter estimation
キーワード：森林、愛媛のスギ、持続的利用、成長モデル、パラメータ推定【原稿受付︰2023 年 7 月 25 日

受理・採録決定︰2023 年 8 月 4 日】

Abstract
Information on forest growth is important for the sustainable use of forests and maximising their value,such as by increasing carbon fixation. This study aims to tentatively estimate a growth model for forests
using data on Japanese cedar forests in Ehime Prefecture by reviewing previous studies on how to estimate
and use a growth model for forests.

要旨
森林の持続的な活用や炭素固定量を増やすなど、森林の価値を最大限発揮させるような利用のために重要となるのは、森林の成長量に関する情報である。本研究では、愛媛県のスギの林分データから、林分の成長モデルをいかに推定するのか、当該モデルをいかに利用するのかについて先行研究をレビューしつつ、林分の成長モデルを仮推定することを目的とする。

1. Importance of estimating forest growthmodelsIn recent years, forest management plans havebeen developed to increase the overall value offorests. This value is not limited to only the wood
supply for housing and wood products but also
includes the use of forest biomass as an energy
source and a sink for carbon dioxide (CO2) to aid indecarbonisation. Moreover, the importance of forest
ecosystem conservation measures as ‘nature-based
solutions’ (IUCN, 2023) to so-called environmental

and social challenges such as disaster resilienceis also emphasised. For the sustainable andenvironmentally sound use of forests, it is necessary
to know the exact rate of forest growth in terms age.
This is because, for example, differences in thinningtiming, thinning location, and thinning intensity
may result in different growth rates, which are
related to carbon fixation, degree of ecosystemconservation, and other factors. Therefore, amodel that can predict forest growth is required tooptimise forest use.

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『愛媛大学社会共創学部紀要』第７巻第２号

However, in numerous cases, the forest growthmodels have not yet been updated. This study aims
to tentatively estimate a growth model for the forest
based on Japanese cedar forest data in the Ehime
Prefecture, referring to Yoshimoto et al. (2012),
while reviewing the literature on how to update thegrowth model and use it for forest management.
2. Forest growth data and existing models
Data on forest growth have traditionally included forest
registers, forest yield tables, and stand density management
diagrams, which were prepared for each prefecture andother entities. A stand density management diagram is amodel used to describe forest growth (Ando, 1966). Thismodel is based on height (tree height), average volume(volume), and the decreasing curve of the number of treesthat die because of natural decay over time (Ando, 1966;
Yoshimoto et al., 2012).
This study estimates a forest growth model by adding
the concept of time to the model (Yoshimoto et al., 2012).
This study uses data from Toyota’s (2013) real forest yield
table for Japanese cedar previously measured and estimatedin Ehime Prefecture forests (Table 1) and parameters fromthe stand density management diagram at Shikoku (Table 2).
Because the parameters of the stand density management
diagram were studied from 1978 to 1988, this study updated
the information on these parameters.
The structure of the stand density management diagram
is as follows (Yoshimoto et al., 2012). The coefficients aredetermined based on region and tree species; p1–p12 are theparameters. Given the number of timber trees per hectare(number of standing trees) N and the height H, the averagevolume per tree v (N, H ) is given by Equation (1.1):

By multiplying (1.1) by N, the number of standing trees
yields Equation (1.2) for the lumber volume per hectare.

The number of standing trees and the height of the trees
are used to calculate the value in Equation (1.3), which isthe forest-shaped height.

2023

Assuming a cylinder with this forest-shaped height as
its height and considering the lumber volume per hectare
as the volume of the cylinder, the cross- sectional area perhectare can be calculated using Equation (1.4).

Furthermore, using Equation (1.4) and N, the average
diameter of the cross-sectional area per tree can be obtained
as in Equation (1.5).

The average diameter of the cross-sectional area is thenadjusted by N and H to obtain the average breast height
diameter per tree as in Equation (1.6).

Although not estimated in the present study, therelationship between N and the number of trees N 0 at thetime of planting is calculated using the natural mortalityline as in Equation (1.7).

3. Methodologies
The estimation methodology in this study followed that ofYoshimoto et al. (2012). The estimation flow is as follows:
first, a height growth model was estimated; then, based on
the estimated height and number of trees, the estimatedvalues using the stand density management diagram werecompared with the values in the forest yield table, and theparameters of the height growth model and stand densitymanagement diagram were re-estimated to best fit themeasured data. The tree height growth model was estimatedusing the Chapman–Richards Equation (1.8) as a trial. All
the parameters were exponentially transformed such thatthey were expected to be positive.

First, the parameters (α, β, γ) of the height growth curve－ 14 －

Growth model of forests as a basis for climate change mitigation and nature-based solutions

in Equation (1.8) were estimated by minimising the sumof residual squares. Using the estimated parameters andthe parameters of the stand density management diagram
as initial values, a tentative parameter that minimises thesum of residual squares as shown in (1.9) was obtained.
From this, the covariance matrix of the residuals wasestimated using (1.10), and the generalised least squareestimator (GLSE)
that minimises the weighted residualsum of squares was estimated as shown in (1.11).

4. Estimation results and discussion
Figure 1 shows graphs of the fitted and original data
(in the forest yield table) of the estimated growth model.
The thin circles represent the original data, and the dottedlines represent the estimated values. The latter reflects theoriginal data very well.
This study tentatively estimated the parameters of thegrowth model and the stand density management diagramsof the ‘Shikoku National Forest Cedar’ created in the years
1978–1988, based on the data of the forest yield table ofciders with the site as type ‘1/5’ in Ehime Prefecture by
Toyota (2013). The parameters can be estimated with thedata of ciders with the site as type ‘2/5’ to ‘5/5’ in the sameway. Although the tentatively estimated parameter values
must be confirmed in the future, this method enabled us tore-estimate the parameters of the growth model and standdensity management diagrams.
The real forest yield table in Toyota (2013) was createdbased on 334 stand structures of Japanese cedar forestsin Ehime Prefecture, rejecting outliers. Because the datain question are for Ehime Prefecture as a whole, if datain specific municipalities are collected in the same way
and parameters are estimated, the tailor-made parametersin those specific areas can be estimated. For example, it
is possible to use Bayesian statistics to re-estimate theposterior distribution of parameters by adding new data,using currently estimated parameters as prior information(Yoshimoto et al., 2012).

Such growth model updates could lead to uses that
increase the overall forest value. For example, by updating
the growth model for a given forest area in a given region,
the amount of carbon that can be fixed from the forests inthat area can be estimated more accurately. If the growthmodels of different forest areas can be compared, it will
be possible to explore the factors that influence the growthof forest areas, such as thinning practices and other socioenvironmental factors, which will be useful in formulating
future management policies. Furthermore, dynamicplanning models can be created based on growth models(Yoshimoto, 2003).
Although this study examined a useful model for forest
unit management, future studies should examine optimal
management methods for forest parcels or multiple forest
units with additional constraints, such as conservation sites(Konoshima and Yoshimoto, 2013). Forest management
for nature-based solutions is also an interesting topic. For
example, it would be meaningful to study the optimisationof forest conditions (number of trees, height, breast
height, diameter, etc.) as well as forest-level optimisationby considering spatial information such as adjacency
and slope for disaster management (Yohimoto, personalcommunication, 1 July 2023).

Acknowledgement
The authors would like to express their greatappreciation to Professor Atsushi Yoshimoto, The
Institute of Statistical Mathematics, for his valuable
and constructive suggestions during the planningand development of this research. His willingness togenerously give his time is greatly appreciated.

References
Ando, T. (1966) Forest Density Management Diagram), Tokyo,
Japan: Norin syuppan. (In Japanese)
Forest Research and Management Organization (2023) YieldRatio Number Ry Calculation Program (shuryo hisu Rykeisan puroguramu), https://www.ffpri.affrc.go.jp/database/yieldindex/ index.html (accessed on 2 July 2023). (InJapanese)
International Union for Conservation of Nature and Natural－ 15 －Resources (IUCN) (2023) Nature-based Solutions, https://『愛媛大学社会共創学部紀要』第７巻第２号
www.iucn.org/our-work/nature-based-solutions (accessedon 2 July 2023).
Konoshima, M., Yoshimoto, A. (2013) Optimization model forforest resource management and its application - harvest
scheduling problems, Proceedings of the Institute of
Statistical Mathematics, 61 (1), 97-109. (In Japanese)
Toyota, N. (2013) On the preparation of forest yield tables forplantation and real forests of Japanese cedar and Japanesecypress in Ehime prefecture (Ehime-ken sugi・hinoki
jinkorin genjitsu rimbunshukakuhyo no sakusei ni tsuite),
Research Report of the Forestry Research Center, Ehime
Prefectural Institute of Agriculture, Forestry and Fisheries(Ehime-ken norin-suisan-kenkyujo ringyo-kenkyu-senta
kenkyu hokoku, 30, 17-34. (In Japanese)
Yohimoto, A. (2003) A dynamic programming model for forest
stand management using MSPATH algorithm, Proceedingsof the Institute of Statistical Mathematics, 51 (1), 73-94. (InJapanese)
Yoshimoto, A., Kamo, K., Yanagihara, H. (2012) Statisticalanalysis of environmental data by R (R ni yoru kankyo datano tokei bunseki ～ Shinrin bunya de no oyo ～ ), Tokyo,
Japan: Asakura syoten. (In Japanese)－ 16 －2023

Growth model of forests as a basis for climate change mitigation and nature-based solutions


Tables and Figures
Table 1. Real forest yield: Japanese cedar site as type ‘1/5’
Adjustment in October 2009㻌

㻌Siteforestindexage㻌㻌㻌Height ofAverage

upper tiertreetrees

height㻌㻌
Total

Average

Numbercross-diameterof trees

sectional
area

Total
trunk
volumeCurrent

Averageannualincrement

increment

24.60

5

12.1

10.2

12.1

2,863

34.27

215.6

43.1

24.60

10

14.3

12.5

15.5

1,946

38.09

267.0

10.3

26.7

24.60

15

16.3

14.5

18.8

1,437

41.39

315.8

9.8

21.1

24.60

20

18.2

16.5

21.9

1,124

44.27

361.9

9.2

18.1

24.60

25

20.0

18.3

25.0

916

46.82

405.3

8.7

16.2

24.60

30

21.6

19.9

27.9

770

49.10

446.2

8.2

14.9

24.60

35

23.2

21.5

30.7663

51.14

484.7

7.7

13.8

24.60

40

24.6

23.0

33.3

582

52.99

520.8

7.2

13.0

24.60

45

25.9

24.3

35.9

520

54.67

554.7

6.8

12.3

24.60

50

27.2

25.6

38.2

470

56.20

586.5

6.4

11.7

24.60

55

28.4

26.8

40.5

430

57.59

616.3

6.0

11.2

24.60

60

29.5

27.9

42.6

397

58.87

644.2

5.6

10.7

24.60

65

30.5

29.0

44.6

369

60.04

670.4

5.2

10.3

24.60

70

31.4

29.9

46.5

346

61.11

694.9

4.9

9.9

24.60

75

32.3

30.8

48.2

326

62.10

717.8

4.6

9.624.60

80

33.1

31.7

49.9

309

63.01

739.2

4.3

9.2

24.60

85

33.9

32.5

51.4

295

63.85

759.3

4.0

8.9

24.60

90

34.6

33.2

52.9

282

64.62

778.0

3.8

8.6

24.60

95

35.3

33.9

54.3

271

65.34

795.6

3.5

8.4

24.60

100

35.9

34.5

55.6

261

66.00

812.0

3.3

8.1

Reference Toyota (2013:31). Appendix 9. Yield table for a real forest area (Japanese cedar 1).

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『愛媛大学社会共創学部紀要』第７巻第２号

2023

Table 2. Parameters of the stand density management diagram of ciders in national forests in Shikoku.

FormulaParameter

Trunk lumber

1

volume per tree
Formula (1.1)

Parameter

Parameter

FormulaParameter

0.0805

Average diameter

80.2529

21.4295

of chest height90.0000

35644.2

Formula (1.6)

100.9813

42.9858

Shape height of a52.1726

forest division60.1453

Formula (1.3)

70.3004



Value





Value





Forest Research and Management Organization (2023); Yoshimoto et al. (2012).
Note: Parameter values of the national cider forests in Shikoku from the Forest Research and Management
Organization (2023) were used.

Average height (m)Age (year)－ 18 －

Growth model of forests as a basis for climate change mitigation and nature-based solutions

Average breast height diameter (cm)

Age (year)

Average lumber volume (m3)

Age (year)
Figure 1. Graphs of fitted and original data (tree height, diameter at breast height, and lumber volume).
Circles indicate original data and dotted lines indicate estimated values.

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